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Saharon Shelah

Publications and source records attributed to Saharon Shelah.

At least 91 records · Page 5Linked to original sources
arXiv2023

AEC for strictly stable

Good frames were suggested in [Sh:h] as the (bare-bones) parallel, in the context of AECs, to superstable (among elementary classes). Here we consider $(μ,λ,κ)$-frames as candidates for being (in the context of AECs) the correct parallel to the class of $|T|^+$-saturated models of a strictly stable theory (among elementary classes). One thing we lose compared to the superstable case is that going up by induction on cardinals is problematic (for stages of small cofinality). But this arises only when we try to lift such classes to higher cardinals. Also, we may use, as a replacement, the existence of prime models over unions of increasing chains. For this context we investigate the dimension.

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arXiv2023

Super black box (formerly: Middle diamond)

This is a slightly corrected version of an old work. Under certain cardinal arithmetic assumptions, we prove that for every large enough regular $λ$ cardinal, for many regular $κ< λ$, many stationary subsets of $λ$ concentrating on cofinality $κ$ have super BB. In particular, we have the super BB on $\{δ< λ\colon cf(δ) = κ\}$. This is a strong negation of uniformization. We have added some details. Works continuing it are [Sh:898] and [Sh:1028]. We thank Ari Brodski and Adi Jarden for their helpful comments. In this paper we had earlier used the notion ``middle diamond" which is now replaced by ``super BB'', that is, ``super black box'', in order to be consistent with other papers (see [Sh:898]).

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arXiv2023

Black Boxes

We shall deal comprehensively with Black Boxes, the intention being that provably in ZFC we have a sequence of guesses of extra structure on small subsets, where the guesses are pairwise almost disjoint; by this we mean they have quite little interaction, and are far apart but together are dense. We first deal with the simplest case, where the existence comes from winning a game by just writing down the opponent's moves. We show how it helps when instead of orders we have trees with boundedly many levels, having freedom in the last. After this we quite systematically look at existence of black boxes, and make connection to non-saturation of natural ideals and diamonds on them.

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arXiv2023

Borel sets with large squares

This is a slightly corrected version of an old work. For a cardinal $μ$ we give a sufficient condition $\oplus_μ$ (involving ranks measuring existence of independent sets) for: $\otimes_μ$ if a Borel set $B\subseteq \mathbb{R} \times \mathbb{R}$ contains a $μ$-square (i.e. a set of the form $A \times A$, with $|A| =μ)$ then it contains a $2^{\aleph_0}$-square and even a perfect square. And also for $\otimes'_μ$ if $ψ\in L_{ω_1, ω}$ has a model of cardinality $μ$ then it has a model of cardinality continuum generated in a ``nice", ``absolute" way. Assuming $\mathrm{MA}+ 2^{\aleph_0}>μ$ for transparency, those three conditions ($\oplus_μ,\otimes_μ$ and $\otimes'e_μ$) are equivalent, and by this we get e.g. $\bigwedge\limits_{α< ω_1} [2^{\aleph_0} \ge \aleph_α\Rightarrow \neg \otimes_{\aleph_α}$], and also $\min\{μ: \otimes_μ\}$ has cofinality $\aleph_1$ if it is $<2^{\aleph_0}$. We deal also with Borel rectangles and related model theoretic problems.

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arXiv2023

The monadic theory of order

We deal with the monadic (second-order) theory of order. We prove all known results in a unified way, show a general way of reduction, prove more results and show the limitation on extending them. We prove (CH) that the monadic theory of the real order is undecidable. Our methods are model-theoretic, and we do not use automaton theory. This is a slightly corrected version of a very old work.

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arXiv2023

Positive logics

Lindström's Theorem characterizes first order logic as the maximal logic satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. If we do not assume that logics are closed under negation, there is an obvious extension of first order logic with the two model theoretic properties mentioned, namely existential second order logic. We show that existential second order logic has a whole family of proper extensions satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. Furthermore, we show that in the context of negationless logics, positive logics, as we call them, there is no strongest extension of first order logic with the Compactness Theorem and the Downward Löwenheim-Skolem Theorem.

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arXiv2023

Stable frames and weights

Was paper 839 in the author's list until winter 2023 when it was divided into three. Part I: We would like to generalize imaginary elements, weight of ortp$(a,M,N), {\mathbf P}$-weight, ${\mathbf P}$-simple types, etc. from [She90, Ch. III,V,\S4] to the context of good frames. This requires allowing the vocabulary to have predicates and function symbols of infinite arity, but it seemed that we do not suffer any real loss. Part II: Good frames were suggested in [She09d] as the (bare bones) right parallel among a.e.c. to superstable (among elementary classes). Here we consider $(μ, λ, κ)$-frames as candidates for being the right parallel to the class of $|T|^+$-saturated models of a stable theory (among elementary classes). A loss as compared to the superstable case is that going up by induction on cardinals is problematic (for cardinals of small cofinality). But this arises only when we try to lift. For this context we investigate the dimension. Part III: In the context of Part II, we consider the main gap problem for the parallel of somewhat saturated model; showing we are not worse than in the first order case.

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arXiv2023

Categoricity and multidimensional diagrams

We study multidimensional diagrams in independent amalgamation in the framework of abstract elementary classes (AECs). We use them to prove the eventual categoricity conjecture for AECs, assuming a large cardinal axiom. More precisely, we show assuming the existence of a proper class of strongly compact cardinals that an AEC which has a single model of some high-enough cardinality will have a single model in any high-enough cardinal. Assuming a weak version of the generalized continuum hypothesis, we also establish the eventual categoricity conjecture for AECs with amalgamation.

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arXiv2023

Superclub, splitting, separating statements

We prove that superclub implies $\mathfrak{s}=\aleph_1$. More generally, superclub at a successor of a weakly compact cardinal implies $\mathfrak{s}_κ=κ^+$. Based on this statement, we separate tiltan from superclub at a successor of a supercompact cardinal. We use Galvin's property in order to separate tiltan from superclub at successors of both regular and singular cardinals.

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arXiv2023

Canonical universal locally finite groups

We prove that for lambda = beta_omega or just lambda strong limit singular of cofinality aleph_0, if there is a universal member in the class K^lf_lambda of locally finite groups of cardinality lambda, then there is a canonical one (parallel to special models for elementary classes, which is the replacement of universal homogeneous ones and saturated ones in cardinals lambda = lambda^ < lambda). For this, we rely on the existence of enough indecomposable such groups, as proved in "Density of indecomposable locally finite groups". We also more generally deal with the existence of universal members in general classes for such cardinals.

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arXiv2023

Torsion-Free Abelian Groups are Borel Complete

We prove that the Borel space of torsion-free Abelian groups with domain $ω$ is Borel complete, i.e., the isomorphism relation on this Borel space is as complicated as possible, as an isomorphism relation. This solves a long-standing open problem in descriptive set theory, which dates back to the seminal paper on Borel reducibility of Friedman and Stanley from 1989.

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arXiv2023

On the weak Borel chromatic number and cardinal invariants of the continuum

We prove that consistently, cov($\mathcal{M})< λ_\mathbf{0} < λ_\mathbf{1} < λ_\mathbf{\infty} < 2^{\aleph_0}$, where $λ_\mathbf{0}$ denotes the weak Borel chromatic number of the Kechris-Solecki-Todorčević graph $\mathbb{G}_0$, that is, the minimal cardinality of a $\mathbb{G}_0$-independent Borel covering of $2^ω$, while $λ_\mathbf{1}$ and $λ_\infty$ are the corresponding invariants of the graph $\mathbb{G}_1$ and the simple graph associated with the equivalence relation $E_0$.

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arXiv2023

Classification over a predicate -- the general case. Part I -- structure theory

We begin a systematic development of structure theory for a first order theory, which is stable over a monadic predicate. We show that stability over a predicate implies quantifier free definability of types over stable sets, introduce an independence notion and explore its properties, prove stable amalgamation results, and show that every type over a model, orthogonal to the predicate, is generically stable.

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arXiv2023

Four Cardinals and Their Relations in ZF

For a set $M$, $\operatorname{fin}(M)$ denotes the set of all finite subsets of $M$, $M^2$ denotes the Cartesian product $M\times M$, $[M]^2$ denotes the set of all $2$-element subsets of $M$, and $\operatorname{seq}^{1-1}(M)$ denotes the set of all finite sequences without repetition which can be formed with elements of $M$. Furthermore, for a set $S$, let $|S|$ denote the cardinality of $S$. Under the assumption that the four cardinalities $|[M]^2|$, $|M^2|$, $|\operatorname{fin}(M)|$, $|\operatorname{seq}^{1-1}(M)|$ are pairwise distinct and pairwise comparable in ZF, there are six possible linear orderings between these four cardinalities. We show that at least five of the six possible linear orderings are consistent with ZF.

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arXiv2023

On slow minimal reals I

Answering a question of Harrington, we show that there exists a proper forcing notion, which adds a minimal real $η\in \prod_{i<ω} n^*_i$, which is eventually different from any old real in $\prod_{i<ω} n^*_i$, where the sequence $\langle n^*_i \mid i<ω\rangle$ grows slowly.

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arXiv2023

On the classification of definable ccc forcing notions

We show that for a Suslin ccc forcing notion $\mathbb Q$ adding a Hechler real, ``$\text{ZF}+\text{DC}_{ω_1}+$all sets of reals are $I_{\mathbb Q,\aleph_0}$-measurable'' implies the existence of an inner model with a measurable cardinal. We also introduce a wide class of Suslin ccc forcing notions which add a Hechler real, so that the above result applies to them.

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arXiv2023

The Halpern--Läuchli Theorem at singular cardinals and failures of weak versions

This paper continues a line of investigation of the Halpern--Läuchli Theorem at uncountable cardinals. We prove in ZFC that the Halpern--Läuchli Theorem for one tree of height $κ$ holds whenever $κ$ is strongly inaccessible and the coloring takes less than $κ$ colors. We prove consistency of the Halpern--Läuchli Theorem for finitely many trees of height $κ$, where $κ$ is a strong limit cardinal of countable cofinality. On the other hand, we prove failure of weak forms of Halpern--\Lauchli\ for trees of height $κ$, whenever $κ$ is a strongly inaccessible, non-Mahlo cardinal or a singular strong limit cardinal with cofinality the successor of a regular cardinal. We also prove failure in $L$ of a weak version for all strongly inaccessible, non-weakly compact cardinals.

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